Sequence of 5 Consecutive Non-Primable Numbers by Changing 1 Digit
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Theorem
The following sequence of $5$ consecutive positive integers cannot be made into prime numbers by changing just one digit:
- $872\,894, 872\,895, 872\,896, 872\,897, 872\,898$
This sequence is A192545 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Proof
Numbers ending in $0$, $2$, $4$, $6$ and $8$ are not prime because by Divisibility by 2 they are divisible by $2$.
Numbers ending in $0$ and $5$ are not prime because by Divisibility by 5 they are divisible by $5$.
Hence each of $872\,894$, $872\,895$, $872\,896$ and $872\,898$ remain composite when you change any of their digits except the last one.
So we inspect the prime factors of the following, only bothering to check the numbers ending in $1$, $3$, $7$ and $9$:
| \(\ds 872\,891\) | \(=\) | \(\ds 271 \times 3221\) | Not prime | |||||||||||
| \(\ds 872\,893\) | \(=\) | \(\ds 7 \times 124699\) | Not prime | |||||||||||
| \(\ds 872\,897\) | \(=\) | \(\ds 263 \times 3319\) | Not prime | |||||||||||
| \(\ds 872\,897\) | \(=\) | \(\ds 17 \times 51347\) | Not prime |
All we need to do now is to inspect $872\,897$.
| \(\ds 072\,897\) | \(=\) | \(\ds 3 \times 11 \times 472\) | Not prime | |||||||||||
| \(\ds 172\,897\) | \(=\) | \(\ds 41 \times 4217\) | Not prime | |||||||||||
| \(\ds 272\,897\) | \(=\) | \(\ds 19 \times 53 \times 271\) | Not prime | |||||||||||
| \(\ds 372\,897\) | \(=\) | \(\ds 3^3 \times 7 \times 1973\) | Not prime | |||||||||||
| \(\ds 472\,897\) | \(=\) | \(\ds 37 \times 12781\) | Not prime | |||||||||||
| \(\ds 572\,897\) | \(=\) | \(\ds 13 \times 127 \times 347\) | Not prime | |||||||||||
| \(\ds 672\,897\) | \(=\) | \(\ds 3 \times 224299\) | Not prime | |||||||||||
| \(\ds 772\,897\) | \(=\) | \(\ds 757 \times 1021\) | Not prime | |||||||||||
| \(\ds 972\,897\) | \(=\) | \(\ds 3 \times 324299\) | Not prime |
| \(\ds 802\,897\) | \(=\) | \(\ds 53 \times 15149\) | Not prime | |||||||||||
| \(\ds 812\,897\) | \(=\) | \(\ds 733 \times 1109\) | Not prime | |||||||||||
| \(\ds 822\,897\) | \(=\) | \(\ds 3^2 \times 91433\) | Not prime | |||||||||||
| \(\ds 832\,897\) | \(=\) | \(\ds 13 \times 79 \times 811\) | Not prime | |||||||||||
| \(\ds 842\,897\) | \(=\) | \(\ds 11 \times 19 \times 37 \times 109\) | Not prime | |||||||||||
| \(\ds 852\,897\) | \(=\) | \(\ds 3 \times 107 \times 2657\) | Not prime | |||||||||||
| \(\ds 862\,897\) | \(=\) | \(\ds 7 \times 131 \times 941\) | Not prime | |||||||||||
| \(\ds 882\,897\) | \(=\) | \(\ds 3 \times 151 \times 1949\) | Not prime | |||||||||||
| \(\ds 892\,897\) | \(=\) | \(\ds 607 \times 1471\) | Not prime |
| \(\ds 870\,897\) | \(=\) | \(\ds 3 \times 61 \times 4759\) | Not prime | |||||||||||
| \(\ds 871\,897\) | \(=\) | \(\ds 13 \times 47 \times 1427\) | Not prime | |||||||||||
| \(\ds 873\,897\) | \(=\) | \(\ds 3 \times 291299\) | Not prime | |||||||||||
| \(\ds 874\,897\) | \(=\) | \(\ds 23 \times 38039\) | Not prime | |||||||||||
| \(\ds 875\,897\) | \(=\) | \(\ds 11 \times 79627\) | Not prime | |||||||||||
| \(\ds 876\,897\) | \(=\) | \(\ds 3^2 \times 7 \times 31 \times 449\) | Not prime | |||||||||||
| \(\ds 877\,897\) | \(=\) | \(\ds 17 \times 113 \times 457\) | Not prime | |||||||||||
| \(\ds 878\,897\) | \(=\) | \(\ds 139 \times 6323\) | Not prime | |||||||||||
| \(\ds 879\,897\) | \(=\) | \(\ds 3 \times 37 \times 7927\) | Not prime |
| \(\ds 872\,097\) | \(=\) | \(\ds 3 \times 149 \times 1951\) | Not prime | |||||||||||
| \(\ds 872\,197\) | \(=\) | \(\ds 59 \times 14783\) | Not prime | |||||||||||
| \(\ds 872\,297\) | \(=\) | \(\ds 191 \times 4567\) | Not prime | |||||||||||
| \(\ds 872\,397\) | \(=\) | \(\ds 3^3 \times 79 \times 409\) | Not prime | |||||||||||
| \(\ds 872\,497\) | \(=\) | \(\ds 37 \times 23581\) | Not prime | |||||||||||
| \(\ds 872\,597\) | \(=\) | \(\ds 11 \times 23 \times 3449\) | Not prime | |||||||||||
| \(\ds 872\,697\) | \(=\) | \(\ds 3 \times 7 \times 29 \times 1433\) | Not prime | |||||||||||
| \(\ds 872\,797\) | \(=\) | \(\ds 17 \times 51341\) | Not prime | |||||||||||
| \(\ds 872\,997\) | \(=\) | \(\ds 3 \times 290999\) | Not prime |
| \(\ds 872\,807\) | \(=\) | \(\ds 13 \times 67139\) | Not prime | |||||||||||
| \(\ds 872\,817\) | \(=\) | \(\ds 3 \times 11 \times 26449\) | Not prime | |||||||||||
| \(\ds 872\,827\) | \(=\) | \(\ds 23 \times 137 \times 277\) | Not prime | |||||||||||
| \(\ds 872\,837\) | \(=\) | \(\ds 7^2 \times 47 \times 379\) | Not prime | |||||||||||
| \(\ds 872\,847\) | \(=\) | \(\ds 3^2 \times 293 \times 331\) | Not prime | |||||||||||
| \(\ds 872\,857\) | \(=\) | \(\ds 43 \times 53 \times 383\) | Not prime | |||||||||||
| \(\ds 872\,867\) | \(=\) | \(\ds 31 \times 37 \times 761\) | Not prime | |||||||||||
| \(\ds 872\,877\) | \(=\) | \(\ds 3 \times 290959\) | Not prime | |||||||||||
| \(\ds 872\,887\) | \(=\) | \(\ds 523 \times 1669\) | Not prime |
The result has been proven.
$\blacksquare$
Sources
- May 1979: Harry L. Nelson: Solutions: 378: II (Crux Mathematicorum Vol. 5: p. 148)
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $872,894$